Advances in Applied Mathematics最新文献

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k}=\left(\begin{array}{c}n\\ k\end{array}\right)n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $\left[n\right]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\varphi )$ in infinitely many indeterminates that give a weight y to each improper edge and a weight $m!{\varphi }_m$ for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

Tree-child networks are one of the most prominent network classes for modeling evolutionary processes which contain reticulation events. Several recent studies have addressed counting questions for bicombining tree-child networks in which every reticulation node has exactly two parents. We extend these studies to d-combining tree-child networks where every reticulation node has now $d\ge 2$ parents, and we study one-component as well as general tree-child networks. For the number of one-component networks, we derive an exact formula from which asymptotic results follow that contain a stretched exponential for $d=2$, yet not for $d\ge 3$. For general networks, we find a novel encoding by words which leads to a recurrence for their numbers. From this recurrence, we derive asymptotic results which show the appearance of a stretched exponential for all $d\ge 2$. Moreover, we also give results on the distribution of shape parameters (e.g., number of reticulation nodes, Sackin index) of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves. We show phase transitions depending on d, leading to normal, Bessel, Poisson, and degenerate distributions. Some of our results are new even in the bicombining case.

Recently Cheng et al. (2023) [7] generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such a major index, namely, the corresponding maj and inv statistics are equidistributed, and exhibit a Haglund-Remmel-Wilson type identity. We then interpret some Jacobi-Rogers polynomials in terms of Laguerre digraphs generalizing Deb and Sokal's alternating Laguerre digraph interpretation of some special Jacobi-Rogers polynomials.

In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on n vertices without k edge-disjoint cycles. This problem had been solved for $k\le 4$. As pointed out by Bollobás, it is very difficult for general k. Recently, Tait and Tobin [J. Comb. Theory, Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of n-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without k edge-disjoint cycles. We first determine the maximum number of edges in a planar graph of order n and maximum degree $n-1$ without k edge-disjoint cycles. Based on this, we then determine the maximum spectral radius as well as its unique extremal graph over all planar graphs on n vertices without k edge-disjoint cycles. Finally, we also discuss several extremal problems for general graphs.

We present a complete classification of $\mathrm{SL}\left(n\right)$ contravariant, $C(R^n\setminus \{o\left\}\right)$-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author Li (2020) [10] which have a good connection with the $L_p$ and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of $\mathrm{SL}\left(n\right)$ contravariant symmetric-tensor-valued valuations on polytopes.

Given $t\ge 2$ and $0\le k\le t$, we prove that the number of labelled k-connected chordal graphs with n vertices and tree-width at most t is asymptotically $c{n}^{-5/2}{\gamma }^{n}n!$, as $n\to \infty$, for some constants $c,\gamma >0$ depending on t and k. Additionally, we show that the number of i-cliques ($2\le i\le t$) in a uniform random k-connected chordal graph with tree-width at most t is normally distributed as $n\to \infty$.

The asymptotic enumeration of graphs of tree-width at most t is wide open for $t\ge 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald (1985) [21], were an algorithm is developed to obtain the exact number of labelled chordal graphs on n vertices.

The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.

We consider the following card guessing game with no feedback. An ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards. One after another a single card is drawn from the top, the guesser makes a guess without seeing the card and gets no response if the guess was correct or not. Building upon and improving earlier results, we provide a limit law for the number of correct guesses and also show convergence of the integer moments.

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence $a{d}^{0}M,adM,a{d}^{2}M,\dots$ of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.